### Problem set 3

```Advanced Microeconomic Theory
Fall 2014
Hannu Vartiainen
Problem Set 3
1. Argue that, in the Second Wefare Theorem, if the consumers’preferences are strictly convex, the Pareto-e¢ cient x is the unique Walrasian
allocation that results from the initial endowment x:
2. Consider n agent exchange economy with n consumers with identical
and strictly concave utility functions. Let there be some initial endowment vector !: Show that equal division is a Pareto-e¢ cient allocation.
3. Consider one consumer, one …rm economy (Robinson Crusoe) where the
consumer’s utility function is of the Cobb-Douglas form u(x; `) = x`;
where x is is the consumption good and ` is the amount of leisure.
The …rm’s production function is linear x = y ; where y is the labor
inputand 2 (0; 1). The resource constraint on the amount of leisure
and labor is ` + y = 1: The consumer owns the …rm.
(a) Write down the consumer’s and the …rms optimization problems,
given the prices p of the output and w of the labor input.
(b) Find the demand function of the consumer and the production
function of the …rm (as a function of p and w)
(c) Solve the Walrasian equilibrium prices p and w
(d) Compute the equilibrium utility of the consumer. What happens
to the utility when changes?
4. Suppose that in the previous exercise, the consumer is an entrepreneur who participates a large market, and adjusts her production and
consumption to the exogenously given prices p0 and w0 . Show that he
utility does not decrease relative to the Robinson Crusoe case.
5. Let X = fx1 ; x2 ; x3 g be the set of pure outcomes where vNM preferences % satisfy x1
x2
x3 . A lottery 1k o¤ers xk with certainty.
Show that there is 2 [0; 1] such that 12
11 + (1
) 13 :
1
6. Let a risk averse agent with Bernoulli utility function u( ) decide his
insurance coverage x > 0. The probability of an accident is
> 0
and the corresponding montery loss is L: Coverage x re‡ects the compensation from the insuance company in the case of an accident. Let
the constant marginal cost of insurance be exactly : What coverage
should the agent choose?
7. By using the formal de…nition of risk-aversion, show that an agent is
risk-averse has a strictly concave Bernoulli utility function.
8. Argue that if a risk averse decision maker rejects a …xed favorable bet,
i.e. one whose expected value is larger than the cost of participating
the bet, at all levels of initial wealth, then the Bernoulli utility of the
decision maker is bounded from above.
2
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